This application is for a renewal of our previous grant entitled "Mathematical Models in Medicine and Biology." Our major interest remains with the analysis of a variety of population genetic models and some elated data analysis especially the interpretation of observed data in human and other populations. The emphasis is both on a more penetrating and complete solution of some of the important classical models and making progress on the more complex models (employing a combination of mathemmtical and computing methodology) with the objective to interpret observed data and gain insight into general population genetic structure and mechanisms. The complexity of the models often precludes any complete analytical treatment and requires the development of appropriate approximation procedures and, in some cases, systematic numerical surveys of the models using computer techniques and simulation. It is expected that many of the models will be tested on actual field data. The interplay between theoretical analysis and an awareness of the biological problems has been a ky factor in our program. The collaborative efforts between the Genetics Department (Prof. L. Cavalli-Sforza since 1971), the Biology Department (Prof. M. Feldman), and the Mathematics Department (Professors S. Karlin and J. McGregor) has provided an ideal set up in achieving several of the research objectives of the Grant. It will be our special aim to maintain and strengthen the unique collaboration we have developed between the Departments of Mathematics, Genetics and Biology. Professor W. Bodmer of Oxford who is closely associated with our group will continue as aregular consultant. A more detailed description of our antcipated research areas follows. In this connection it should be clear that our interdisciplinry collaboration encourages the study of problems as they arise in continuing research activities.